Learning Contractive Integral Operators with Fredholm Integral Neural Operators

2026-04-03

AI Summary

This work addresses the challenge of learning non-expansive integral operators arising in high-dimensional Fredholm integral equations of the second kind by proposing Fredholm Integral Neural Operators (FREDINOs). FREDINOs constitute the first neural operator architecture that simultaneously guarantees universal approximation capability and strict contractivity, thereby ensuring that the learned solution operator satisfies the contraction mapping condition and enabling provable convergence of fixed-point iterations. By integrating Fredholm theory, neural operator design, and boundary integral equation frameworks, the method efficiently solves both linear and nonlinear integral equations and extends naturally to high-dimensional nonlinear elliptic partial differential equations. Numerical experiments on multidimensional benchmark problems demonstrate that FREDINOs achieve high accuracy, strong interpretability, and excellent generalization performance.

Abstract

We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral Neural Operators (FREDINOs), for FIEs and prove that they are universal approximators of linear and non-linear integral operators and corresponding solution operators. We furthermore prove that the learned operators are guaranteed to be contractive, thereby strictly satisfying the mathematical property required for the convergence of the fixed point scheme. Finally, we also demonstrate how FREDINOs can be used to learn the solution operator of non-linear elliptic PDEs, via a Boundary Integral Equation (BIE) formulation. We assess the proposed methodology numerically, via several benchmark problems: linear and non-linear FIEs in arbitrary dimensions, as well as a non-linear elliptic PDE in 2D. Built on tailored mathematical/numerical analysis theory, FREDINOs offer high-accuracy approximations and interpretable schemes, making them well suited for scientific machine learning/numerical analysis computations.

Problem

• Fredholm Integral Equations
• contractive operators
• integral operators
• solution operators
• nonlinear elliptic PDEs

Innovation

• Fredholm Integral Neural Operators
• contractive operators
• universal approximation
• Boundary Integral Equation
• scientific machine learning